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doi: 10.3934/krm.2022015
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Thermalization of a rarefied gas with total energy conservation: Existence, hypocoercivity, macroscopic limit

1. 

University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

2. 

Würzburg University, Department of mathematics, Emil Fischer Str. 40, 97074 Würzburg, Germany

*Corresponding author: Gianluca Favre

Received  September 2021 Revised  March 2022 Early access May 2022

Fund Project: This work has been supported by the Austrian Science Fund, grants no. W1245 and F65, and by the Humboldt foundation

The thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law.

For the coupled nonlinear system of the kinetic and the heat equation, existence of solutions is proved on the one-dimensional torus. Spectral stability of the equilibrium is shown on the torus in arbitrary dimensions by hypocoercivity methods. The macroscopic limit towards a nonlinear cross-diffusion problem is carried out formally.

Citation: Gianluca Favre, Marlies Pirner, Christian Schmeiser. Thermalization of a rarefied gas with total energy conservation: Existence, hypocoercivity, macroscopic limit. Kinetic and Related Models, doi: 10.3934/krm.2022015
References:
[1]

F. AchleitnerA. Arnold and D. Stüerzer, Large-time behaviour in non-symmetric Fokker-Planck equations, Riv. Math. Univ. Parma, 6 (2015), 1-68. 

[2]

A. Arnold and J. Erb, Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, preprint, 2014, arXiv: 1409.5425.

[3]

M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 124 (2006), 881-912.  doi: 10.1007/s10955-005-8075-x.

[4]

R. Bosi and M. Cáceres, The BGK model with external confining potential: Existence, long-time behaviour and time-periodic Maxwellian equilibria, J. Stat. Phys., 136 (2009), 297-330.  doi: 10.1007/s10955-009-9782-5.

[5]

Y.-P. Choi and S.-B. Yun, Global existence of weak solutions for Navier-Stokes-BGK system, Nonlinearity, 33 (2020), 1925-1955.  doi: 10.1088/1361-6544/ab6c38.

[6]

S. De BiévreT. Goudon and A. Vavasseur, Particles interacting with a vibrating medium: Existence of solutions and convergence to the Vlasov-Poisson system, SIAM J. Math. Anal., 48 (2016), 3984-4020.  doi: 10.1137/16M1065306.

[7]

J. DolbeaultA. KlarC. Mouhot and C. Schmeiser, Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes, Appl. Math. Res. Express, 2013 (2013), 165-175.  doi: 10.1093/amrx/abs015.

[8]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc, 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.

[9]

G. Favre, A. Jüngel, C. Schmeiser and N. Zamponi, Existence analysis of a degenerate diffusion system for heat-conducting gases, Nonlinear Diff. Equ. Appl. NoDEA, 28 (2021), Paper No. 41, 28 pp. doi: 10.1007/s00030-021-00700-z.

[10]

G. Favre, M. Pirner and C. Schmeiser, Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model, preprint, 2022, arXiv: 2012.15622.

[11]

G. Favre and C. Schmeiser, Hypocoercivity and fast reaction limit for linear reaction networks with kinetic transport, J. Stat. Phys., 178 (2020), 1319-1335.  doi: 10.1007/s10955-020-02503-5.

[12]

F. Golse and L. Saint-Raymond, Hydrodynamic limits for the Boltzmann equation, Riv. Mat. Univ. Parma, 2 (2005), 1-144. 

[13]

J. HaskovecS. HittmeirP. Markowich and A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, SIAM J. Math. Anal., 50 (2018), 1037-1075.  doi: 10.1137/16M1062065.

[14]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359. 

[15]

M. Herda and L. Rodrigues, Large-time behavior of solutions to Vlasov-Poisson-Fokker-Planck equations: From evanescent collisions to diffusive limit, J. Stat. Phys., 170 (2018), 895-931.  doi: 10.1007/s10955-018-1963-7.

[16]

S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.  doi: 10.1007/BF03167846.

[17]

C. Klingenberg and M. Pirner, Existence, uniqueness and positivity of solutions for BGK models for mixtures, J. Diff. Equ., 264 (2018), 702-727.  doi: 10.1016/j.jde.2017.09.019.

[18]

L. Liu and M. Pirner, Hypocoercivity for a BGK model for gas mixtures, J. Diff. Equ., 267 (2019), 119-149.  doi: 10.1016/j.jde.2019.01.006.

[19]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.

[20]

S. Park and S.-B. Yun, Cauchy problem for the ellipsoidal BGK model for polyatomic particles, J. Diff. Equ., 266 (2019), 7678-7708.  doi: 10.1016/j.jde.2018.12.013.

[21]

B. Perthame, Global existence to the BGK model of the Boltzmann equation, J. Diff. Equ., 82 (1989), 191-205.  doi: 10.1016/0022-0396(89)90173-3.

[22]

B. Perthame and M. Pulvirenti, Weighted $L^\infty$-bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal., 125 (1993), 289-295.  doi: 10.1007/BF00383223.

[23]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., (2009).  doi: 10.1090/S0065-9266-09-00567-5.

[24]

S.-B. Yun, Classical solutions for the ellipsoidal BGK model with fixed collision frequency, J. Diff. Equ., 259 (2015), 6009-6037.  doi: 10.1016/j.jde.2015.07.016.

show all references

References:
[1]

F. AchleitnerA. Arnold and D. Stüerzer, Large-time behaviour in non-symmetric Fokker-Planck equations, Riv. Math. Univ. Parma, 6 (2015), 1-68. 

[2]

A. Arnold and J. Erb, Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, preprint, 2014, arXiv: 1409.5425.

[3]

M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 124 (2006), 881-912.  doi: 10.1007/s10955-005-8075-x.

[4]

R. Bosi and M. Cáceres, The BGK model with external confining potential: Existence, long-time behaviour and time-periodic Maxwellian equilibria, J. Stat. Phys., 136 (2009), 297-330.  doi: 10.1007/s10955-009-9782-5.

[5]

Y.-P. Choi and S.-B. Yun, Global existence of weak solutions for Navier-Stokes-BGK system, Nonlinearity, 33 (2020), 1925-1955.  doi: 10.1088/1361-6544/ab6c38.

[6]

S. De BiévreT. Goudon and A. Vavasseur, Particles interacting with a vibrating medium: Existence of solutions and convergence to the Vlasov-Poisson system, SIAM J. Math. Anal., 48 (2016), 3984-4020.  doi: 10.1137/16M1065306.

[7]

J. DolbeaultA. KlarC. Mouhot and C. Schmeiser, Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes, Appl. Math. Res. Express, 2013 (2013), 165-175.  doi: 10.1093/amrx/abs015.

[8]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc, 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.

[9]

G. Favre, A. Jüngel, C. Schmeiser and N. Zamponi, Existence analysis of a degenerate diffusion system for heat-conducting gases, Nonlinear Diff. Equ. Appl. NoDEA, 28 (2021), Paper No. 41, 28 pp. doi: 10.1007/s00030-021-00700-z.

[10]

G. Favre, M. Pirner and C. Schmeiser, Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model, preprint, 2022, arXiv: 2012.15622.

[11]

G. Favre and C. Schmeiser, Hypocoercivity and fast reaction limit for linear reaction networks with kinetic transport, J. Stat. Phys., 178 (2020), 1319-1335.  doi: 10.1007/s10955-020-02503-5.

[12]

F. Golse and L. Saint-Raymond, Hydrodynamic limits for the Boltzmann equation, Riv. Mat. Univ. Parma, 2 (2005), 1-144. 

[13]

J. HaskovecS. HittmeirP. Markowich and A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, SIAM J. Math. Anal., 50 (2018), 1037-1075.  doi: 10.1137/16M1062065.

[14]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359. 

[15]

M. Herda and L. Rodrigues, Large-time behavior of solutions to Vlasov-Poisson-Fokker-Planck equations: From evanescent collisions to diffusive limit, J. Stat. Phys., 170 (2018), 895-931.  doi: 10.1007/s10955-018-1963-7.

[16]

S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.  doi: 10.1007/BF03167846.

[17]

C. Klingenberg and M. Pirner, Existence, uniqueness and positivity of solutions for BGK models for mixtures, J. Diff. Equ., 264 (2018), 702-727.  doi: 10.1016/j.jde.2017.09.019.

[18]

L. Liu and M. Pirner, Hypocoercivity for a BGK model for gas mixtures, J. Diff. Equ., 267 (2019), 119-149.  doi: 10.1016/j.jde.2019.01.006.

[19]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.

[20]

S. Park and S.-B. Yun, Cauchy problem for the ellipsoidal BGK model for polyatomic particles, J. Diff. Equ., 266 (2019), 7678-7708.  doi: 10.1016/j.jde.2018.12.013.

[21]

B. Perthame, Global existence to the BGK model of the Boltzmann equation, J. Diff. Equ., 82 (1989), 191-205.  doi: 10.1016/0022-0396(89)90173-3.

[22]

B. Perthame and M. Pulvirenti, Weighted $L^\infty$-bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal., 125 (1993), 289-295.  doi: 10.1007/BF00383223.

[23]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., (2009).  doi: 10.1090/S0065-9266-09-00567-5.

[24]

S.-B. Yun, Classical solutions for the ellipsoidal BGK model with fixed collision frequency, J. Diff. Equ., 259 (2015), 6009-6037.  doi: 10.1016/j.jde.2015.07.016.

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